➕ Algebra 1 · Linear Functions
Standard Form of a Line
Write a line as Ax + By = C, find both intercepts by covering a term, graph from them, switch to y = mx + b, and model totals of two quantities.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- A-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-REI.D.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
- F-IF.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
By the end you can…
- Name A, B and C in an equation written as Ax + By = C, and write an equation in the usual standard form (integers, A not negative).
- Find the x-intercept and y-intercept of a line from its standard form, and graph the line from them.
- Convert between standard form and slope-intercept form, including results with negative or fractional numbers.
- Recognize horizontal and vertical lines in standard form, and write an equation for a total made of two quantities.
Lesson (school style)
Standard form writes the equation of a line as Ax + By = C. Both variables sit on the left, and a plain number sits on the right. It hides the slope, but it is very good at two jobs: finding both intercepts in seconds, and describing situations where two amounts add up to a fixed total. It can also write every line, including vertical ones.
The form Ax + By = C
In Ax + By = C, the letters A, B and C stand for fixed numbers, and x and y are the variables. The x term and the y term sit on the left, and the number C sits on the right. A and B cannot both be 0.
For 3x + 2y = 12, A = 3, B = 2 and C = 12. Each number keeps its sign: in 2x − 5y = 10, B = −5, because 2x − 5y means 2x + (−5)y.
The usual convention. Many classes ask for the final answer in a tidy version where:
- A, B and C are integers (no fractions or decimals),
- A is not negative, and
- A, B and C have no common factor except 1.
This is a convention, not a law of math. The equation 6x + 4y = 24 is still an equation of the same line as 3x + 2y = 12, because dividing every term by 2 does not change which points make it true. The convention just picks one tidy version so that answers are easy to compare. If your teacher or book uses a different version, follow theirs.
Solutions and the graph
A solution of 3x + 2y = 12 is a pair (x, y) that makes the equation true. For example, (2, 3) is a solution because 3(2) + 2(3) = 6 + 6 = 12. But (2, 4) is not, because 6 + 8 = 14.
The graph of the equation is all of its solutions plotted on the coordinate plane. When A and B are not both 0, the graph of Ax + By = C is a straight line. To test a point, substitute its coordinates. If both sides match, the point is on the line.
Find both intercepts: cover a term
Every point on the x-axis has y = 0, and every point on the y-axis has x = 0. So you get each intercept by making the other variable 0, which you can picture as covering its term.
- Cover the y-term. What is left is Ax = C, so x = C/A (assume A ≠ 0). The line crosses the x-axis at (C/A, 0), so the x-intercept is C/A.
- Cover the x-term. What is left is By = C, so y = C/B (assume B ≠ 0). The line crosses the y-axis at (0, C/B), so the y-intercept is C/B.
For 3x + 2y = 12: covering 2y leaves 3x = 12, so x = 4 and the line crosses at (4, 0). Covering 3x leaves 2y = 12, so y = 6 and the line crosses at (0, 6).
Keep each sign with its number. In 3x − 10y = 30, covering 3x leaves −10y = 30, so y = −3, not 3.
“Covering” is only a picture of setting a variable to 0: 0 times anything is 0, so that term disappears.
One catch: if C = 0 and neither A nor B is 0, as in 2x + 3y = 0, both intercepts are 0, so the line crosses both axes at the same point, (0, 0). That gives only one point, so find another one with a table of values or the slope.
Graph from the intercepts
- Find both intercepts by covering terms.
- Plot the two points, (C/A, 0) and (0, C/B).
- Draw a straight line through them and extend it in both directions with arrowheads.
- Check with a third point by substituting it into the equation.
This works best when the intercepts are easy to plot, such as whole numbers. If they are awkward, as in 4x + 7y = 5 (intercepts 5/4 and 5/7), find points another way, such as a table of values. And if C = 0, the two intercepts are the same point, so you need a second point from somewhere else. If A = 0 or B = 0, the line is horizontal or vertical and crosses only one axis, so use the method in “Horizontal and vertical lines” below.
From standard form to slope-intercept form
Solve for y, as you would for a literal equation. Subtract Ax from both sides, then divide every term by B:
By = −Ax + C, so y = −(A/B)x + C/B (assume B ≠ 0, because you divide by it).
So the slope is m = −A/B and the y-intercept is b = C/B. Notice the minus sign in the slope: for 3x + 2y = 12, the slope is −3/2, not 3/2. These shortcuts are handy, but it is safer to show the steps. When B is negative, dividing by B changes the sign of every term.
From other forms to standard form
To write a line in standard form, get both variable terms on the left and the number on the right, then tidy up.
- Clear fractions. Multiply every term by the common denominator.
- Move the terms. Add or subtract so that the x and y terms are on the left side and the plain number is on the right.
- Tidy up if you want the usual convention. If A is negative, multiply every term by −1 (it is not 0, so the line does not change). If A, B and C share a common factor, divide every term by it.
The same steps work from point-slope form: distribute first, then move the terms (see worked example 7).
Horizontal and vertical lines
Standard form can write every line.
- If A = 0, the equation is By = C, so y = C/B. This is a horizontal line with slope 0. For example, 6y = 18 is y = 3.
- If B = 0, the equation is Ax = C, so x = C/A. This is a vertical line with an undefined slope. For example, 8x = −24 is x = −3.
A vertical line cannot be written as y = mx + b, but it fits standard form. The slope shortcut −A/B also fails for it, because B = 0 would mean dividing by 0. That agrees with the undefined slope.
A horizontal line y = c with c ≠ 0 never meets the x-axis, so it has no x-intercept. A vertical line x = c with c ≠ 0 never meets the y-axis, so it has no y-intercept.
Standard form in real life
Standard form is natural when two quantities add up to a total. If each item has a price, then price × number gives the money for that item, and the terms add up to the total spent.
Define each variable in words first. Say adult tickets cost 8 dollars and student tickets cost 5 dollars. Let a be the number of adult tickets and s the number of student tickets. Total sales of 120 dollars give 8a + 5s = 120.
- The intercepts are the extreme cases where everything is one kind: all adult tickets or all student tickets.
- The slope −A/B is the trade-off: when the total stays fixed, each 1 more of the x item changes the y item by −A/B (a negative change means fewer).
- Counts of items must be whole numbers and cannot be negative, so only some points on the line (in the first quadrant) fit the story.
Key words
- Standard form
- A way to write a line’s equation as Ax + By = C, with the variable terms on the left and a number on the right.
- Coefficient
- The number multiplying a variable. In 3x + 2y = 12, the coefficients are 3 and 2, and each keeps its sign.
- Constant
- A number on its own, with no variable. In Ax + By = C, it is C.
- x-intercept
- The x-value where a line crosses the x-axis. If that value is a, the line crosses at (a, 0). For Ax + By = C with A ≠ 0, it is C/A.
- y-intercept
- The y-value where a line crosses the y-axis. For Ax + By = C with B ≠ 0, it is C/B, and the line crosses at the point (0, C/B).
- Integer
- A number with no fraction or decimal part. The integers are …, −2, −1, 0, 1, 2, …
- Convention
- An agreed habit that makes work easy to compare. It is not a law of math.
Worked examples
Example 1. Find both intercepts of 4x + 3y = 24 by covering a term, then graph the line.
- Cover the y-term. What is left is 4x = 24, so x = 6. The line crosses the x-axis at (6, 0), so the x-intercept is 6.
- Cover the x-term. What is left is 3y = 24, so y = 8. The line crosses the y-axis at (0, 8), so the y-intercept is 8.
- Plot (6, 0) and (0, 8) and draw a straight line through them, with arrowheads at both ends.
- Check with a third point, (3, 4): 4(3) + 3(4) = 12 + 12 = 24, so it is on the line.
- The line goes down from left to right. The slope is −A/B = −4/3, which agrees: from (0, 8), go right 3 and down 4 to reach (3, 4).
Answer: Answer: A line through (6, 0) and (0, 8). The x-intercept is 6 and the y-intercept is 8.
Example 2. Find both intercepts of 2x − 7y = 14 and graph the line.
- Cover the y-term. What is left is 2x = 14, so x = 7. The line crosses the x-axis at (7, 0).
- Cover the x-term. What is left is −7y = 14. Divide both sides by −7: y = −2. The line crosses the y-axis at (0, −2). The minus sign stays with the 7, because B = −7 and C/B = 14 ÷ (−7) = −2.
- Plot (7, 0) and (0, −2) and draw the line. It goes up from left to right, which agrees with the slope −A/B = −2/(−7) = 2/7.
- Check with a third point to the left, (−7, −4): 2(−7) − 7(−4) = −14 + 28 = 14. It is on the line.
Answer: Answer: A line through (7, 0) and (0, −2). The x-intercept is 7 and the y-intercept is −2.
Example 3. Write 5x − 4y = 20 in slope-intercept form.
- Subtract 5x from both sides: −4y = −5x + 20.
- Divide every term by −4: y = (−5x) ÷ (−4) + 20 ÷ (−4) = (5/4)x − 5.
- So m = 5/4 and b = −5.
- Compare with the shortcuts: m = −A/B = −5/(−4) = 5/4 and b = C/B = 20/(−4) = −5. They match.
- Check with x = 4: y = (5/4)(4) − 5 = 0, and 5(4) − 4(0) = 20. It matches.
Answer: Answer: y = (5/4)x − 5, with slope 5/4 and y-intercept −5.
Example 4. Write each equation in standard form with integers and A not negative: (a) y = −(3/4)x + 5/2, (b) y = (1/2)x + 3.
- (a) The denominators are 4 and 2. Multiply every term by 4: 4y = −3x + 10.
- (a) Add 3x to both sides: 3x + 4y = 10. Here A = 3 is not negative, and 3, 4 and 10 have no common factor. Check with the point (2, 1), which is on the original line since −(3/4)(2) + 5/2 = 1: 3(2) + 4(1) = 10.
- (b) Multiply every term by 2: 2y = x + 6. Subtract x from both sides: −x + 2y = 6.
- (b) Here A = −1 is negative. Multiply every term by −1 (not 0, so the line stays the same): x − 2y = −6.
- (b) Check with the point (4, 5), which is on the original line since (1/2)(4) + 3 = 5: 4 − 2(5) = −6.
Answer: Answer: (a) 3x + 4y = 10. (b) x − 2y = −6.
Example 5. Describe the graphs of 6y = 18 and 8x = −24.
- In 6y = 18, the x term is missing, so A = 0. Divide by 6: y = 3. The graph is a horizontal line through (0, 3), with slope 0. It has no x-intercept.
- In 8x = −24, the y term is missing, so B = 0. Divide by 8: x = −3. The graph is a vertical line through (−3, 0), with an undefined slope. It has no y-intercept, and it cannot be written as y = mx + b.
- The slope shortcut −A/B would give −8/0 for the second line, which is not a number. That agrees with the undefined slope.
Answer: Answer: 6y = 18 is the horizontal line y = 3. 8x = −24 is the vertical line x = −3.
Example 6. A school play sells adult tickets for 8 dollars and student tickets for 5 dollars. One night the ticket sales total exactly 120 dollars. Let a be the number of adult tickets and s the number of student tickets. Write an equation, find both intercepts and say what they mean, and find which whole-number combinations are possible.
- Each ticket type brings in price × number: 8a dollars from adults and 5s dollars from students. Together that is 120 dollars, so 8a + 5s = 120. This is standard form with A = 8, B = 5 and C = 120.
- Put a on the horizontal axis and s on the vertical axis. Cover the s-term: 8a = 120, so a = 15. The point (15, 0) means 15 adult tickets and no student tickets.
- Cover the a-term: 5s = 120, so s = 24. The point (0, 24) means 24 student tickets and no adult tickets.
- Suppose 10 adult tickets were sold: 8(10) + 5s = 120, so 5s = 40 and s = 8. The point (10, 8) is on the line.
- Could exactly 7 adult tickets be sold? 8(7) + 5s = 120 gives 5s = 64, so s = 12.8. The point (7, 12.8) is on the line, but 12.8 tickets is not possible.
- Checking the whole numbers from 0 to 15 for a, only a = 0, 5, 10, 15 give a whole number of student tickets: (0, 24), (5, 16), (10, 8) and (15, 0).
- The slope is −A/B = −8/5. In this story, 5 more adult tickets (40 dollars more) go with 8 fewer student tickets (40 dollars less), so the total stays at 120 dollars.
Answer: Answer: 8a + 5s = 120. The line crosses the axes at (15, 0) (all adult tickets) and (0, 24) (all student tickets). The possible whole-number combinations are (0, 24), (5, 16), (10, 8) and (15, 0).
Example 7. Write y + 2 = 3(x − 1) in standard form.
- Distribute the 3: y + 2 = 3x − 3.
- Subtract 3x from both sides to bring the x term to the left, and subtract 2 from both sides to move the number to the right: −3x + y = −5.
- Here A = −3 is negative. Multiply every term by −1: 3x − y = 5.
- Check with the known point (1, −2): 3(1) − (−2) = 3 + 2 = 5. It matches.
Answer: Answer: 3x − y = 5
In plain English
Standard form is a tidy way to write a line: Ax + By = C. The x term and the y term sit together on one side. A plain number sits on the other side.
It is great when two things add up to a total. Say you buy two kinds of snack and spend a fixed amount. The cost of the first kind plus the cost of the second kind equals the total. That is standard form.
It also makes the intercepts easy. To find where the line crosses the x-axis, cover the y term and solve what is left. To find where it crosses the y-axis, cover the x term and solve what is left.
Most classes write the final answer with no fractions or decimals and with a first number that is not negative. The other two numbers can be negative. That is a habit, not a law of math.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Cupcakes and cookies
A club has exactly 30 dollars to spend at a bake sale. Cupcakes cost 3 dollars each and cookies cost 2 dollars each. Let c be the number of cupcakes and k the number of cookies, so 3c + 2k = 30. Buying 6 cupcakes and 6 cookies costs 18 + 12 = 30 dollars. Swap 2 cupcakes (6 dollars) for 3 cookies (6 dollars) and the total stays at 30: 4 cupcakes and 9 cookies cost 12 + 18 = 30 dollars.
How it connects: Each term is price × number, and the terms add up to the fixed total. Covering terms gives the intercepts: all cookies is k = 15, and all cupcakes is c = 10. With c on the horizontal axis, the slope −3/2 is the swap: 2 fewer cupcakes goes with 3 more cookies. The limit: you can only buy whole treats, so only some points on the line fit the story. The point (c, k) = (5, 7.5) is on the line (15 + 15 = 30) but would mean 7.5 cookies.
Counting baskets
In a basketball game, suppose a team scored only from 2-point baskets and 3-point baskets (no free throws) and finished with 36 points. Let x be the number of 2-point baskets and y the number of 3-point baskets, so 2x + 3y = 36. For example, 9 two-point baskets and 6 three-point baskets give 18 + 18 = 36 points.
How it connects: Each term is points per basket × number of baskets, and the terms add up to the total: A = 2, B = 3 and C = 36. Covering terms gives the intercepts: x = 18 (all 2-point baskets) and y = 12 (all 3-point baskets). The limit: counts must be whole numbers, so only some points on the line fit. Real games also have 1-point free throws, which would add a third term, so two variables is a simplified model.
Eggs and baskets
The proverb “Don’t put all your eggs in one basket” warns against risking everything on a single thing. Suppose you split 40 dollars between 5-dollar game passes (x of them) and 4-dollar snack packs (y of them), so 5x + 4y = 40. Buying 4 passes and 5 snack packs costs 20 + 20 = 40 dollars.
How it connects: The intercepts are the all-eggs-in-one-basket cases: x = 8 (only passes) and y = 10 (only snack packs). Every other point on the line between them splits the 40 dollars. The limit: the proverb is about risk, and the math is only about arithmetic. Every point on the line spends the same 40 dollars, and the line does not tell you which split is wiser. Also, only whole-number points can actually be bought.
Two sides of the same coin
To say that two things are “two sides of the same coin” means that they are different ways of looking at one thing.
How it connects: The equations 3x + 2y = 12 and y = −(3/2)x + 6 are two sides of the same coin: the same line and the same points, written differently. Both contain (4, 0) and (0, 6). Standard form shows the intercepts quickly. Slope-intercept form shows the slope and the y-intercept at a glance. The limit: a line has more than two forms (point-slope form is a third), and the forms are not equally flexible, because a vertical line fits standard form but not slope-intercept form.
The potion shop
In a made-up game, a potion costs 4 coins and a shield costs 6 coins. You have exactly 60 coins. Let p be the number of potions and s the number of shields, so 4p + 6s = 60. Dividing every term by 2 gives 2p + 3s = 30. The pair (3, 8) works in both: 12 + 48 = 60 and 6 + 24 = 30.
How it connects: Dividing every term of an equation by the same number, as long as it is not 0, does not change its graph. That is why the convention (no common factor) tidies 4p + 6s = 60 into 2p + 3s = 30. Both have the intercepts p = 15 and s = 10. The limit: the coin story lives in the first equation, where 4p is the coins spent on potions. In the tidy version the numbers are half-prices and half the budget, so read the story from the original equation.
Memory help
Cover y, find x
To find the x-intercept, cover the y term and solve what is left. To find the y-intercept, cover the x term and solve what is left. Say: “Cover y, find x. Cover x, find y.” For 3x + 2y = 12, covering 2y leaves 3x = 12, so x = 4.
WAT is standard?
Three habits for a tidy Ax + By = C: Without fractions (A, B and C are integers), A is not negative, Trimmed (no common factor). These habits are a convention, not a law. The equation 6x + 4y = 24 is still a correct equation of its line. It is just not the tidy version.
Read it like a receipt
When A and B are prices, read Ax + By = C as a receipt: A dollars for each x, plus B dollars for each y, adds up to C dollars in total. Each term is price × number.
The slope is −A/B
In 3x + 2y = 12, the slope is −3/2, not 3/2 and not 3. Solve for y to see why: y = −(3/2)x + 6. The minus sign appears because the x term moves to the other side.
Signs stay with their numbers
In 2x − 5y = 10, B = −5. Covering 2x leaves −5y = 10, so the y-intercept is −2, not 2.
Study notes
Key ideas
- Standard form is Ax + By = C, with A and B not both 0. The graph is a straight line.
- The usual convention is integers for A, B and C, with A not negative and no common factor. It is a convention, not a law of math.
- Cover the y term to find the x-intercept C/A (if A ≠ 0). Cover the x term to find the y-intercept C/B (if B ≠ 0).
- To get slope-intercept form, solve for y. The slope is −A/B and the y-intercept is C/B (if B ≠ 0).
- A = 0 gives a horizontal line. B = 0 gives a vertical line, which standard form can write but slope-intercept form cannot.
- Standard form fits totals of two quantities: price × number for each item, added to a fixed total.
Steps
- To find intercepts: cover the y term and solve for x, then cover the x term and solve for y.
- To graph: plot the two intercept points, draw the line through them, and check a third point.
- To reach slope-intercept form: subtract the x term, then divide every term by B.
- To reach standard form: clear fractions, move the x and y terms to the left and the number to the right, make A not negative, and remove any common factor.
Formulas
- Standard form: Ax + By = C, with A and B not both 0
- x-intercept: x = C/A (if A ≠ 0), so the line crosses at (C/A, 0)
- y-intercept: y = C/B (if B ≠ 0), so the line crosses at (0, C/B)
- Slope: m = −A/B (if B ≠ 0)
- Horizontal line: A = 0: y = C/B, slope 0
- Vertical line: B = 0: x = C/A, slope undefined
Watch out
- Signs belong to their numbers: in 2x − 5y = 10, B = −5.
- The slope is −A/B, with a minus sign. It is not A/B.
- To find the x-intercept, set y = 0 (cover the y term). Setting x = 0 finds the y-intercept.
- When you divide by a negative B, every term changes sign. Divide every term, not just one.
- If C = 0, both intercepts are 0 and the line crosses both axes at (0, 0), so you need another point.
Why it works
- Why cover a term? The x-axis has y = 0, so the y term becomes 0 and disappears. The y-axis has x = 0, so the x term disappears.
- Why is the slope −A/B? Solving Ax + By = C for y gives y = −(A/B)x + C/B, and the number multiplying x is the slope.
- Why can you multiply every term by a number that is not 0? Doing the same thing to both sides keeps the same solutions, so the graph does not change. Multiplying by 0 would destroy the equation.
Common mistakes
Teach it back
Explain to a friend how to graph 6x + 5y = 30 using the intercepts, how to write it in slope-intercept form, and how to check that (−5, 12) is on the line.
Explaining it out loud shows what you really know. Did you…
- I covered the y term to get 6x = 30, so the x-intercept is 5 and the line crosses at (5, 0). I covered the x term to get 5y = 30, so the y-intercept is 6 and the line crosses at (0, 6).
- I subtracted 6x and divided every term by 5 to get y = −(6/5)x + 6, so the slope is −6/5.
- I checked (−5, 12) by substituting: 6(−5) + 5(12) = −30 + 60 = 30, so it is on the line.
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
Downloads a calendar file (.ics). Works with Google, Apple and Outlook calendars. Nothing is sent to us.